Topological superconductors Ismail Achmed-Zade - - PowerPoint PPT Presentation

Topological superconductors

Ismail Achmed-Zade

Arnold-Sommerfeld-Center

December 14th, 2017

Ismail Achmed-Zade (ASC) Topological superconductors December 14th, 2017 1 / 24

Overview

1

Superconductivity BCS theory

2

Topology

3

Topological Superconductor Bulk-boundary Correspondence

4

Experimental results

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What is a topological superconductor?

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BCS theory

Idea

An electron pair interacts via a phonon

• k
• k −

q

• q
• k′
• k′ +

q

Recall

Phonons are the quanta of background lattice oscillations.

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BCS theory

Procedure

Start with the Hamiltonian describing electrons coupling to the background lattice H = H0 + Hint. with H0 =

• q ω

qb†

• qb

q +

• k ǫ

ka†

• ka

k

Hint =

• q,

k M qa†

• k+

qa k(b q + b† − q).

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BCS theory

Frohlichers Idea

Using the operator S = b†

− q

ǫ

k − ǫ k+ q − ω q

+ b

q

ǫ

k + ǫ k+ q − ω q

• M

qa†

• q+

ka k,

we make a unitary transformation H′ = e−SHeS, which yields H′ = H0 +

• k,k′,q

|M

q|2a† k+qa† k′−qakak′

ωq (ǫk′ − ǫk′−q)2 − (ωq)2

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BCS theory

Computation shows that electrons form Cooper pairs with opposite spin Bogoljubov approach ak,↑a−k,↓ = 0 Condensate of Cooper pairs

Effective Hamiltonian

H =

• ǫσ1,σ2(k)a†

k,σ1ak,σ2 +

1 2

• Vσ1,...,σ4(k, k′)a†

k,σ1a† −k,σ2ak′,σ3a−k′,σ4

∼

• ǫσ1,σ2(k)a†

k,σ1ak,σ2 + 1

2

• ∆σ1,σ2(k)a†

k,σ1a† −k,σ2 + h.c.

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A calculation for later

Rewrite H = (a†

k,σ1, a−k,σ1)

ǫσ1,σ2(k) ∆σ1,σ2(k) ∆†

σ1,σ2(k)

−ǫT

σ1,σ2(−k)

ak,σ2 a†

−k,σ2

• In the 2D case we can assume normalized eigenstates of the form

|u(kx, ky) = cos(2α(kx, ky)) sin(2α(kx, ky))

• ,

where α is a certain function of ǫ(k) and ∆(k).

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Bogoljubov quasi particles

Particle-hole duality allows diagonalization of H, yielding Bogoljubov modes UHU† =

• Eiγ†

k,σγk,σ,

with γk↑ = ukak↑ − vka†

−k↓

γk↓ = ukak↓ + vka†

−k↑.

Important point: Possibility for Majorana excitations γ†

0 = γ0.

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A little bit of math

What is topology?

Roughly topology is the study of shapes. It is one of the ’craziest’ parts of mathematics. We are going to study vector bundles over smooth manifolds and their characteristic classes, or rather the first Chern class c1 and the first Chern number

• cn

1 .

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The Berry Connection

Consider the Brillouin zone of an effectively 2D solid kx ky Hkx,ky ψ(kx, ky)

Berry Connection

• A(

k) = iψ( k)|∇

kψ(

k) A − → A − ∇φ Fij = ǫij∂iAj

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The Berry Connection

First Chern number

ν = 1 2π

• 2DBZ

dA = 1 2π

• 2DBZ

ǫij∂iAjdkxdky =

• 2DBZ

F 2π ∈ Z. Appearance of field strength F Thm: c1(L) = F/2π For 2D insulators σxy = − e2

h ν

Time reversal: A(k) → A(−k) implies F(k) → −F(−k)

• F(k) →
• −F(−k) = −
• F(k) ⇒ ν = 0.

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Closer look at time reversal

Kramers rule

Consider a fermion in 2+1 dimensions Tψ(t, x1, x2) = iσ1ψ(−t, x1, x2) ⇒ T 2 = −1. Further by anti-unitarity of T, i.e. u|v = Tv|Tu u|Tu = T 2u|Tu = −u|Tu. Thus u|Tu = 0, and we get a 2-fold degenerate energy levels.

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Closer look at time reversal

Now we write a basis for our Hilbert space at momentum (kx, ky) Hkx,ky = spanu1(k), eiφ1Tu1(−k), .... We obtain to different Berry connections per energy level AI

n(k) and

AII

n (k). Then

2πν =

• n
• F I

n + F II n = νI + νII = 0,

by T symmetry. However νI = −νII ⇒ (−1)νI = (−1)νII ∈ Z2

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Topological numbers

ky kx ky = const γ

1D Example

Take a one-dimensional slice in the 2D Brillouin zone of the SCD example at the beginning S1 → S1 kx → cos(2α(kx)) sin(2α(kx))

• Ismail Achmed-Zade (ASC)

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Winding number

This map gives a winding number as ω = 1 2π π

−π

∂kx(2α(kx)) ∼

• kx:ǫ(kx)=0

sign(d(k))sign(∂kxǫ(kx)) Because the path γ is homotopy equivalent to ky = const, the corresponding gapless mode has a flat energy dispersion.

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Gapless modes on the boundary

ν1 ν2 For gapped systems and En < EF the following map is continuous BZ − → H

• k

→ ψn( k) This implies that the Chern number is constant. How then do we model discontinuities? By allowing bands to intersect at the boundary. This gives rise to gapless modes.

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Gapless modes on the boundary

By allowing bands to intersect at the boundary. This gives rise to gapless modes.

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1D quantum wire

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Thank you

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Boundary Theory

The boundary of a 3+1 D topological superconductor admits a Majorana fermion IEuclidean =

• Y

¯ ψiDψd3x, where Dαβ = ǫαγDγ

β.

The partition function is given by the Pfaffian Zψ = Pf (D). In the case of D =

• D

−DT

• ,

we have Pf (D) = detD.

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Kramers rule once more

There is again a Kramers rule: If Dχ = λχ, then ˜ χα = (T χ)α = ǫαβχ∗

β is also an eigenvector

D˜ χ = λ˜ χ. Using [D, T ] = 0 this implies D = ⊕i     −λi λi −λi λi     ⇒ Pf (D) = Π′λi

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An anomaly?

Fermionic anomalies do not affect the absolute value of Zψ = |Zψ|eiφ Time reversal sends Zψ → ¯ Zψ Zψ ∈ R+ implies no anomalies We would like to check the sign of Pf (D) = Π′λi. Consider a family of these theories differing only by gauge transformations → apply index theory of the (lifted) Dirac operator (on the mapping torus) (g0, A0) φ · (g0, A0) Y Y φ(Y ) It turns out that the index on a mapping torus of this form is always identically zero!!

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But...

If we regularize the Pfaffian and then we obtain a complex partition function Zψ = |Zψ| exp(i π 4 η). But considering the interaction with some bulk theory we can restore T

• invariance. The result however depends on the four manifold X,

Zψ = (−1)IX . I is the index of a Dirac operator on a manifold with boundary.

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